MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , if1(true(), x, y, xs) -> min(x, xs) , if1(false(), x, y, xs) -> min(y, xs) , min(x, cons(y, z)) -> if1(le(x, y), x, y, z) , min(x, nil()) -> x , if2(true(), x, y, xs) -> xs , if2(false(), x, y, xs) -> cons(y, del(x, xs)) , del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) , del(x, nil()) -> nil() , minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) , minsort(nil()) -> nil() } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(0(), 0()) -> c_4() , eq^#(0(), s(y)) -> c_5() , eq^#(s(x), 0()) -> c_6() , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_8(min^#(x, xs)) , if1^#(false(), x, y, xs) -> c_9(min^#(y, xs)) , min^#(x, cons(y, z)) -> c_10(if1^#(le(x, y), x, y, z), le^#(x, y)) , min^#(x, nil()) -> c_11() , if2^#(true(), x, y, xs) -> c_12() , if2^#(false(), x, y, xs) -> c_13(del^#(x, xs)) , del^#(x, cons(y, z)) -> c_14(if2^#(eq(x, y), x, y, z), eq^#(x, y)) , del^#(x, nil()) -> c_15() , minsort^#(cons(x, y)) -> c_16(min^#(x, y), minsort^#(del(min(x, y), cons(x, y))), del^#(min(x, y), cons(x, y)), min^#(x, y)) , minsort^#(nil()) -> c_17() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(0(), 0()) -> c_4() , eq^#(0(), s(y)) -> c_5() , eq^#(s(x), 0()) -> c_6() , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_8(min^#(x, xs)) , if1^#(false(), x, y, xs) -> c_9(min^#(y, xs)) , min^#(x, cons(y, z)) -> c_10(if1^#(le(x, y), x, y, z), le^#(x, y)) , min^#(x, nil()) -> c_11() , if2^#(true(), x, y, xs) -> c_12() , if2^#(false(), x, y, xs) -> c_13(del^#(x, xs)) , del^#(x, cons(y, z)) -> c_14(if2^#(eq(x, y), x, y, z), eq^#(x, y)) , del^#(x, nil()) -> c_15() , minsort^#(cons(x, y)) -> c_16(min^#(x, y), minsort^#(del(min(x, y), cons(x, y))), del^#(min(x, y), cons(x, y)), min^#(x, y)) , minsort^#(nil()) -> c_17() } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , if1(true(), x, y, xs) -> min(x, xs) , if1(false(), x, y, xs) -> min(y, xs) , min(x, cons(y, z)) -> if1(le(x, y), x, y, z) , min(x, nil()) -> x , if2(true(), x, y, xs) -> xs , if2(false(), x, y, xs) -> cons(y, del(x, xs)) , del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) , del(x, nil()) -> nil() , minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) , minsort(nil()) -> nil() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,5,6,11,12,15,17} by applications of Pre({1,2,4,5,6,11,12,15,17}) = {3,7,8,9,10,13,14,16}. Here rules are labeled as follows: DPs: { 1: le^#(0(), y) -> c_1() , 2: le^#(s(x), 0()) -> c_2() , 3: le^#(s(x), s(y)) -> c_3(le^#(x, y)) , 4: eq^#(0(), 0()) -> c_4() , 5: eq^#(0(), s(y)) -> c_5() , 6: eq^#(s(x), 0()) -> c_6() , 7: eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , 8: if1^#(true(), x, y, xs) -> c_8(min^#(x, xs)) , 9: if1^#(false(), x, y, xs) -> c_9(min^#(y, xs)) , 10: min^#(x, cons(y, z)) -> c_10(if1^#(le(x, y), x, y, z), le^#(x, y)) , 11: min^#(x, nil()) -> c_11() , 12: if2^#(true(), x, y, xs) -> c_12() , 13: if2^#(false(), x, y, xs) -> c_13(del^#(x, xs)) , 14: del^#(x, cons(y, z)) -> c_14(if2^#(eq(x, y), x, y, z), eq^#(x, y)) , 15: del^#(x, nil()) -> c_15() , 16: minsort^#(cons(x, y)) -> c_16(min^#(x, y), minsort^#(del(min(x, y), cons(x, y))), del^#(min(x, y), cons(x, y)), min^#(x, y)) , 17: minsort^#(nil()) -> c_17() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_8(min^#(x, xs)) , if1^#(false(), x, y, xs) -> c_9(min^#(y, xs)) , min^#(x, cons(y, z)) -> c_10(if1^#(le(x, y), x, y, z), le^#(x, y)) , if2^#(false(), x, y, xs) -> c_13(del^#(x, xs)) , del^#(x, cons(y, z)) -> c_14(if2^#(eq(x, y), x, y, z), eq^#(x, y)) , minsort^#(cons(x, y)) -> c_16(min^#(x, y), minsort^#(del(min(x, y), cons(x, y))), del^#(min(x, y), cons(x, y)), min^#(x, y)) } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , eq^#(0(), 0()) -> c_4() , eq^#(0(), s(y)) -> c_5() , eq^#(s(x), 0()) -> c_6() , min^#(x, nil()) -> c_11() , if2^#(true(), x, y, xs) -> c_12() , del^#(x, nil()) -> c_15() , minsort^#(nil()) -> c_17() } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , if1(true(), x, y, xs) -> min(x, xs) , if1(false(), x, y, xs) -> min(y, xs) , min(x, cons(y, z)) -> if1(le(x, y), x, y, z) , min(x, nil()) -> x , if2(true(), x, y, xs) -> xs , if2(false(), x, y, xs) -> cons(y, del(x, xs)) , del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) , del(x, nil()) -> nil() , minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) , minsort(nil()) -> nil() } Obligation: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , eq^#(0(), 0()) -> c_4() , eq^#(0(), s(y)) -> c_5() , eq^#(s(x), 0()) -> c_6() , min^#(x, nil()) -> c_11() , if2^#(true(), x, y, xs) -> c_12() , del^#(x, nil()) -> c_15() , minsort^#(nil()) -> c_17() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_8(min^#(x, xs)) , if1^#(false(), x, y, xs) -> c_9(min^#(y, xs)) , min^#(x, cons(y, z)) -> c_10(if1^#(le(x, y), x, y, z), le^#(x, y)) , if2^#(false(), x, y, xs) -> c_13(del^#(x, xs)) , del^#(x, cons(y, z)) -> c_14(if2^#(eq(x, y), x, y, z), eq^#(x, y)) , minsort^#(cons(x, y)) -> c_16(min^#(x, y), minsort^#(del(min(x, y), cons(x, y))), del^#(min(x, y), cons(x, y)), min^#(x, y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , if1(true(), x, y, xs) -> min(x, xs) , if1(false(), x, y, xs) -> min(y, xs) , min(x, cons(y, z)) -> if1(le(x, y), x, y, z) , min(x, nil()) -> x , if2(true(), x, y, xs) -> xs , if2(false(), x, y, xs) -> cons(y, del(x, xs)) , del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) , del(x, nil()) -> nil() , minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y)))) , minsort(nil()) -> nil() } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Weak Usable Rules: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , if1(true(), x, y, xs) -> min(x, xs) , if1(false(), x, y, xs) -> min(y, xs) , min(x, cons(y, z)) -> if1(le(x, y), x, y, z) , min(x, nil()) -> x , if2(true(), x, y, xs) -> xs , if2(false(), x, y, xs) -> cons(y, del(x, xs)) , del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) , del(x, nil()) -> nil() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , eq^#(s(x), s(y)) -> c_7(eq^#(x, y)) , if1^#(true(), x, y, xs) -> c_8(min^#(x, xs)) , if1^#(false(), x, y, xs) -> c_9(min^#(y, xs)) , min^#(x, cons(y, z)) -> c_10(if1^#(le(x, y), x, y, z), le^#(x, y)) , if2^#(false(), x, y, xs) -> c_13(del^#(x, xs)) , del^#(x, cons(y, z)) -> c_14(if2^#(eq(x, y), x, y, z), eq^#(x, y)) , minsort^#(cons(x, y)) -> c_16(min^#(x, y), minsort^#(del(min(x, y), cons(x, y))), del^#(min(x, y), cons(x, y)), min^#(x, y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , eq(0(), 0()) -> true() , eq(0(), s(y)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , if1(true(), x, y, xs) -> min(x, xs) , if1(false(), x, y, xs) -> min(y, xs) , min(x, cons(y, z)) -> if1(le(x, y), x, y, z) , min(x, nil()) -> x , if2(true(), x, y, xs) -> xs , if2(false(), x, y, xs) -> cons(y, del(x, xs)) , del(x, cons(y, z)) -> if2(eq(x, y), x, y, z) , del(x, nil()) -> nil() } Obligation: innermost runtime complexity Answer: MAYBE The input cannot be shown compatible Arrrr..